Density of States and Thouless Formula for Random Unitary Band Matrices

We study the density of states measure for some class of random unitary band matrices and prove a Thouless formula relating it to the associated Lyapunov exponent. This class of random matrices appears in the study of the dynamical stability of certain quantum systems and can be considered as a unitary version of the Anderson model. It is also related with orthogonal polynomials on the unit circle. We further determine the support of the density of states measure and provide a condition ensuring it possesses an analytic density. Communicated by Eugene BogomolnySubmitted 07/11/03, accepted 15/01/04     

一类随机游动上确界的密度的渐近性

Kluppelberg^[1], Asmussen 等^[2] 研究了增量有有限负均值的随机游动上确界的密度的渐近性. 该文则在 Denisov 等^[3], 程东亚和王岳宝^[4]的基础上, 进一步研究了增量均值为负无穷的随机游动上确界的密度的渐近性. 最后, 为了说明常见重尾分布大多满足上述结果的条件, 该文给出了一些分布族的性质.     

Finite Diagonal Random Matrices

The goal of this article is to extend some results of Popescu (Probab. Theory Relat. Fields 144:179, 2009) in several directions. We establish the limiting spectral distribution (LSD) for r-diagonal matrices under reduced moment conditions compared to those required by Popescu. We also deal with the joint convergence of several sequences of such matrices. In particular, we show that there is a large class of such matrices where the joint limit is not free while the marginals are semicircular. We also consider matrices of the form $X_{n}X_{n}^{T}$ where X _n is a sequence of nonsymmetric r-diagonal random matrices and establish their limiting spectral distribution.     

Asymptotics of Maxima of Discrete Random Variables

If X₁, X₂,..., X_n are independent and identically distributed discrete random variables and M_n=max (X₁,..., X_n) we examine the limiting behavior of (M_n–b(n))/a(n) as n . It is well known that for discrete distributions such as Poisson and geometric the limiting distribution is not non-degenerate. However, by tuning the parameters of the discrete distribution to vary as n , it is possible to obtain non-degenerate limits for (M_n–b(n))/a(n). We consider four families of discrete distributions and show how this can be done.     

Products of Random Rectangular Matrices

We study the asymptotic behaviour of points under matrix cocyles generated by rectangular matrices. In particular we prove a random Perron‐Frobenius and a Multiplicative Ergodic Theorem. We also provide an example where such products of random rectangular matrices arise in the theory of random walks in random environments and where the Multiplicative Ergodic Theorem can be used to investigate recurrence problems.     

Asymptotics of Partial Density Functions for Divisors

We study the asymptotic behaviour of the partial density function associated to sections of a positive hermitian line bundle that vanish to a particular order along a fixed divisor Y. Assuming the data in question is invariant under an \(S^1\)-action (locally around Y) we prove that this density function has a distributional asymptotic expansion that is in fact smooth upon passing to a suitable real blow-up. Moreover we recover the existence of the “forbidden region” R on which the density function is exponentially small, and prove that it has an “error-function” behaviour across the boundary \(\partial R\). As an illustrative application, we use this to study a certain natural function that can be associated to a divisor in a Kähler manifold.     

Spectral Measures of Spiked Random Matrices

Journal of Theoretical Probability - We study two spiked models of random matrices under general frameworks corresponding, respectively, to additive deformation of random symmetric matrices and...     

Complex Outliers of Hermitian Random Matrices

In this paper, we study the asymptotic behavior of the outliers of the sum a Hermitian random matrix and a finite rank matrix which is not necessarily Hermitian. We observe several possible convergence rates and outliers locating around their limits at the vertices of regular polygons as in Benaych-Georges and Rochet (Probab Theory Relat Fields, 2015), as well as possible correlations between outliers at macroscopic distance as in Knowles and Yin (Ann Probab 42(5):1980–2031, 2014) and Benaych-Georges and Rochet (2015). We also observe that a single spike can generate several outliers in the spectrum of the deformed model, as already noticed in Benaych-Georges and Nadakuditi (Adv Math 227(1):494–521, 2011) and Belinschi et al. (Outliers in the spectrum of large deformed unitarily invariant models 2012, arXiv:1207.5443v1). In the particular case where the perturbation matrix is Hermitian, our results complete the work of Benaych-Georges et al. (Electron J Probab 16(60):1621–1662, 2011), as we consider fluctuations of outliers lying in “holes” of the limit support, which happen to exhibit surprising correlations.     

Asymptotic Density and the Asymptotics of Partition Functions

Let A be a set of positive integers with gcd (A) = 1, and let p _A (n) be the partition function of A. Let c ₀ = 2/3. If A has lower asymptotic density and upper asymptotic density , then lim inf log p _A (n)/c ₀ n and lim sup log p _A (n)/c ₀ n . In particular, if A has asymptotic density > 0, then log p _A (n) c₀n. Conversely, if > 0 and log p _A (n) c ₀ n, then the set A has asymptotic density .     

Asymptotics for Random Walks with Dependent Heavy-Tailed Increments

We consider a random walk {S _n} with dependent heavy-tailed increments and negative drift. We study the asymptotics for the tail probability P{sup _n S _n >x} as x. If the increments of {S _n} are independent then the exact asymptotic behavior of P{sup _n S _n >x} is well known. We investigate the case in which the increments are given as a one-sided asymptotically stationary linear process. The tail behavior of sup _n S _n turns out to depend heavily on the coefficients of this linear process.     

Circular Law for Noncentral Random Matrices

Let (X _jk )_j,k≥1 be an infinite array of i.i.d. complex random variables with mean 0 and variance 1. Let λ _n,1,…,λ _n,n be the eigenvalues of \((\frac{1}{\sqrt{n}}X_{jk})_{1\leqslant j,k\leqslant n}\). The strong circular law theorem states that, with probability one, the empirical spectral distribution \(\frac{1}{n}(\delta _{\lambda _{n,1}}+\cdots+\delta _{\lambda _{n,n}})\) converges weakly as n→∞ to the uniform law over the unit disc {z∈?,|z|≤1}. In this short paper, we provide an elementary argument that allows us to add a deterministic matrix M to (X _jk )_{1≤ j,k ≤ n} provided that Tr(MM ^*)=O(n ²) and rank(M)=O(n ^α ) with α<1. Conveniently, the argument is similar to the one used for the noncentral version of the Wigner and Marchenko–Pastur theorems.     

Random Matrices and Erasure Robust Frames

Data erasure can often occur in communication. Guarding against erasures involves redundancy in data representation. Mathematically this may be achieved by redundancy through the use of frames. One way to measure the robustness of a frame against erasures is to examine the worst case condition number of the frame with a certain number of vectors erased from the frame. The term numerically erasure-robust frames was introduced in Fickus and Mixon (Linear Algebra Appl 437:1394–1407, 2012) to give a more precise characterization of erasure robustness of frames. In the paper the authors established that random frames whose entries are drawn independently from the standard normal distribution can be robust against up to approximately 15 % erasures, and asked whether there exist frames that are robust against erasures of more than 50 %. In this paper we show that with very high probability random frames are, independent of the dimension, robust against erasures as long as the number of remaining vectors is at least \(1+\delta _0\) times the dimension for some \(\delta _0>0\). This is the best possible result, and it also implies that the proportion of erasures can be arbitrarily close to 1 while still maintaining robustness. Our result depends crucially on a new estimate for the smallest singular value of a rectangular random matrix with independent standard normal entries.     

The Polynomial Method for Random Matrices

We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semicircle law and the Marčenko–Pastur law are special cases. Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue density function. The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form. In this article, we develop the mathematics of the polynomial method which allows us to describe the class of algebraic matrices by its generators and map the constructive approach we employ when proving algebraicity into a software implementation that is available for download in the form of the RMTool random matrix “calculator” package. Our characterization of the closure of algebraic probability distributions under free additive and multiplicative convolution operations allows us to simultaneously establish a framework for computational (noncommutative) “free probability” theory. We hope that the tools developed allow researchers to finally harness the power of infinite random matrix theory.     

Asymptotics of Damped Periodic Motions with Random Initial Speed

We consider motion on the circle, possibly with friction and external forces, the initial velocity being a large random variable. We prove that under various assumptions the probability law of the stopping position of the motion converges to a distribution depending only on the motion equation. Here the time of stopping is either a constant or the first time instant at which the velocity vanishes, and the initial velocity is of the form αU + β, where U is a fixed random variable and α and/or β tend to infinity.     

Random Laplacian Matrices and Convex Relaxations

The largest eigenvalue of a matrix is always larger or equal than its largest diagonal entry. We show that for a class of random Laplacian matrices with independent off-diagonal entries, this bound is essentially tight: the largest eigenvalue is, up to lower order terms, often the size of the largest diagonal. entry. Besides being a simple tool to obtain precise estimates on the largest eigenvalue of a class of random Laplacian matrices, our main result settles a number of open problems related to the tightness of certain convex relaxation-based algorithms. It easily implies the optimality of the semidefinite relaxation approaches to problems such as \({{\mathbb {Z}}}_2\) Synchronization and stochastic block model recovery. Interestingly, this result readily implies the connectivity threshold for Erd?s–Rényi graphs and suggests that these three phenomena are manifestations of the same underlying principle. The main tool is a recent estimate on the spectral norm of matrices with independent entries by van Handel and the author.     

Distribution of Subdominant Eigenvalues of Random Matrices

We mainly investigate the behavior of the subdominant eigenvalue of matrices B= (b _i,j)^n,n whose entries are independent random variables with an expectation Eb _i,j=1/n and with a variance n c/n ² for some constant c 0. For such matrices we show that for large n, the subdominant eigenvalue is, with great probability, in a small neighborhood of 0. We also show that for large n, the spectral radius of such matrices is, with great probability, in a small neighborhood of 1.     

Random Matrices and Complexity of Spin Glasses

We give an asymptotic evaluation of the complexity of spherical p‐spin spin glass models via random matrix theory. This study enables us to obtain detailed information about the bottom of the energy landscape, including the absolute minimum (the ground state), and the other local minima, and describe an interesting layered structure of the low critical values for the Hamiltonians of these models. We also show that our approach allows us to compute the related TAPcomplexity and extend the results known in the physics literature. As an independent tool, we prove a large deviation principle for the k^th‐largest eigenvalue of the Gaussian orthogonal ensemble, extending the results of Ben Arous, Dembo, and Guionnet. © 2012 Wiley Periodicals, Inc.